Since the uncompressed NIDS converges at a linear rate \(\mathcal {O}\big (\max \lbrace \frac{L}{\mu },\frac{1}{\rho }\rbrace \log \frac{1}{\epsilon }\big )\)  [1]}, the \(\delta \) -contracted compression only affects the term related to the network. Particularly, COLD converges at the same rate as NIDS over a network with a “modified" adjacency matrix \(W^{\prime }\)  such that \(\lambda _2(W^{\prime })=1-(1-\delta )^2\lambda _2(W)\) , and reduces to NIDS if \(\delta =0\) . Note that COLD is communication-efficient since each node only sends the compressed vector \(\mathbf {q}_i^k\) .
